# 08/04/2020: Algebraic Topology (Allen Hatcher) /home/zack/Dropbox/Library/Allen Hatcher/Algebraic Topology (609)/Algebraic Topology - Allen Hatcher.pdf Last Annotation: 08/04/2020 ## Highlights - Example 0\.5\. Since RP n is obtained from RPn−1 by attaching an n cell, the infinite union RP∞ can view RP ∞ = S = n RPn becomes a cell complex with one cell in each S ∞ as the space of lines through the origin in R∞ = n R n R (Allen Hatcher 15) - The existence of a retracting homomorphism ρ : G→H is quite a strong condition on H \. If H is a normal subgroup, it implies that G is the direct product of H and the kernel of ρ \. If H is not normal, then G is what is called in group theory the semi-direct product of H and the kernel of ρ \. (Allen Hatcher 45) - Lemma 1\.19\. If ϕt : X →Y is a homotopy and h is the path ϕt \(x0 \) formed by the images of a basepoint x0 ∈ X , then the three maps in the diagram at the right satisfy ϕ0∗ = βh ϕ1∗ \. (Allen Hatcher 46) - Φ : Aα Theorem 1\.20\. If X is the union of path-connected open sets A α each containing the basepoint x0 ∈ X and if each intersection Aα ∩ Aβ is path-connected, then the h homomorphism morphism Φ : ∗α π1 \(Aα \)→π Aβ ∩ Aγ is path-connected, \)→π1 \(X\) is surjective\. If in addition each intersection homomorphism Φ : ∗α π1 \(Aα \)→π1 \(X\) is surjective\. If in addition each intersection Aα ∩ Aβ ∩ Aγ is path-connected, then the kernel of Φ is the normal subgroup N generated by all elements of the form iαβ \(ω\)iβα \(ω\)−1 for ω ∈ π1 \(Aα ∩ Aβ \) , and hence Φ induces an isomorphism π1 \(X\) ≈ ∗α π1 \(Aα \)/N \. Example 1\.21: Wedge Sums\. In Chapter 0 we defined the (Allen Hatcher 52) - a free product of nontrivial groups has trivial center\. (Allen Hatcher 57) - \(a\) If Y is obtained from X by attaching 2 cells as described above, then the inclusion X ֓ Y induces a surjection π1 \(X, x0 \)→π1 \(Y , x0 \) whose kernel is N \. Thus π1 \(Y \) ≈ π1 \(X\)/N \. \(b\) If Y is obtained from X by attaching n cells for a fixed n > 2 , then the inclusion X ֓ Y induces an isomorphism π1 \(X, x0 \) ≈ π1 \(Y , x0 \) \. \(c\) For a path-connected cell complex X the inclusion of the 2 skeleton X 2 ֓ X induces an isomorphism π1 \(X 2 , x0 \) ≈ π1 \(X, x0 \) \. (Allen Hatcher 59) - First we have the homotopy lifting property, also known as the covering homotopy property: (Allen Hatcher 69) - Proposition tion 1\.30\. Given a covering space p p : Xe →X , a homotopy ft : Y →X , and a a unique homotopy fet : Y →X e of fe0 that Propos map fe0 lifts ft \. ition 1\.30\. Given a covering space p : Xe →X , a homoto : Y →Xe lifting f0 , then there exists a unique homotopy f (Allen Hatcher 69) - Taking Y to be a point gives the path lifting property for a covering space p:Xe →X , which says that for each path f : I →X and each lift x e point f \(0\) = x there is a unique path fe : I →X 0 e lifting f starting at x e 0 of t x e \. 0 (Allen Hatcher 69) - Proposition f : \(Y , y0 \)→ y0 \)→\(X, 1\.33\. Suppose given a covering space p : \( X, x0 \) with Y path-connected and locally p : \(X, e xe 0 \)→\(X, x0 \) ally path-connected\. e 0 \) x thand a map Then a lift (Allen Hatcher 70) - : \(Y , y0 When \)→\(X, e xe we say a x e 0 \) of f exists iff f∗ π1 \(Y , y0 \) a space has a certain property  ⊂ p∗ π1 \(X, e x e0 locally, such as  x e0 \) \. as being  \. ing locally pathconnected, we usually mean that each point has arbitrarily small open neighborhoods with this property\. (Allen Hatcher 70) - Thus for Y to be locally path-connected means that for each point (Allen Hatcher 70) - Proposition with X and X and X e p For a loop 1\.32\. The numb e path-connected X op g in X based 2\. The number of s ath-connected equals g in X based at x0 , sheets of a covering space p :  ls the index of p∗ π1 \(X, e xe 0 \) i 0 , let g e x p : \(X, e 0 \)  in π1 \(X, \)→\(X, x0 \) X, x0 \) \. (Allen Hatcher 70) - Proposition p : \(X, e x p : \(X, e consists e x \(X, e nsists e 0 \) x s of ion 1\.31\. The map p∗ : π1 \(X, e 0 \)→ e x \)→\(X, x0 \) is injective\. The image f the homotopy classes of loops in 0 \)→π1 \(X, x0 \) in age subgroup p∗ in X based at x0 p x 0 induced b p∗ π1 \(X, e x0 whose X e s by y a c  x e 0 \) lifts a covering space  in π1 \(X, x0 \) s to X at x e xe 0 \)→\(X, x0 \) is injective\. The image subgroup p∗ π1 \(X, e xe 0 \) in ists of the homotopy classes of loops in X based at x0 whose lifts to (Allen Hatcher 70) - y ∈ Y and each neighborhood U of y there is an open neighborhood V ⊂ U of y that is path-connected\. (Allen Hatcher 71) - Each point x ∈ X has a neighborhood U such that the inclusion-induced map π1 \(U, x\)→π1 \(X, x\) is trivial; one says X is semilocally simply-connected (Allen Hatcher 72) - A necessary condition for X to have a simply-connected covering space (Allen Hatcher 72) - A locally simply-connected space is certainly semilocally simply-connected\. (Allen Hatcher 72) - CW complexes have the much stronger property of being locally contractible, (Allen Hatcher 72) - An example of a space that is not semilocally simplyconnected is the shrinking wedge of circles, the subspace X ⊂ R 2 consisting of the circles of radius 1/n centered at the point \(1/n , 0\) for n = 1, 2, ··· , (Allen Hatcher 72) - if we take the cone CX = \(X × I\)/\(X × {0}\) on the shrinking wedge of circles, this is semilocally simply-connected since it is contractible, but it is not locally simply-connected\. (Allen Hatcher 72) - e is path-connected\. In particular, only dentity deck transformation can fix a point of Xe\. e →X is called normal if for each x ∈ X and each pair of lifts A covering space p : X e\. e →X is called normal if for each x A covering space p : X ′ x, e xe of x there is a deck transformation taking x e to xe ′\. F (Allen Hatcher 79) -  e x π1 \(X, e 0 \) ⊂ This covering x e 0 \) ove  ⊂ π1 \(X, x0 \) \. Then : ring space is normal iff H is a normal subgroup of π1 \(X, x0 \) \. (Allen Hatcher 80) - If X is a space and A is a nonempty closed subspace that is a deformation retract of some neighborhood in X (Allen Hatcher 123) - e n \(X/A\) can be represented by a chain α in X with ∂α a cycle in A se homology class is ∂x ∈ He n−1 \(A\) \. Pairs of spaces \(X, A\) satisfying the hypothesis of the theorem will be called good pairs (Allen Hatcher 123) - For example, if X is a CW complex and A is a nonempty subcomplex, then \(X, A\) is a good pair by Proposition A\.5 in the Appendix\. (Allen Hatcher 123) - If f : S n →S n has no fixed points then deg f = \(−1\)n+1 (Allen Hatcher 143) - If f ≃ g then deg f = deg g since f∗ = g∗ \. The converse statement, that f ≃ g if deg f = deg g , is a fundamental theorem of Hopf from around 1925 which we prove in Corollary 4\.25\. (Allen Hatcher 143) - deg f = −1 if f is a reflection of S n , fixing the points in a subsphere S n and interchanging the two complementary hemispheres (Allen Hatcher 143) - Note that the antipodal map has no fixed points (Allen Hatcher 144) - In the case of S 1 , the map f \(z\) = z k , where we view S 1 as the unit circle in C , has degree k (Allen Hatcher 146) - Proposition 2\.33\. deg Sf = deg f , where Sf : S n+1 →S n+1 is the suspension of the map f : S n →S n \. (Allen Hatcher 146) - A rotation is a homeomorphism so its local degree at any point equals its global degree, (Allen Hatcher 146) - 0 → → Cn \(A ∩ B\) ----ϕ ϕ → Cn \(A\) ⊕ Cn \(B\) ----ψ ψ → Cn \(A + B\) → 0 (Allen Hatcher 158) - if A∩B is path-connected, the H1 terms of the reduced Mayer–Vietoris sequence yield an isomorphism H1 \(X\) ≈ H1 \(A\) statement of the van Kampen theorem, ⊕ H1 \(B\) , and H B\) H  / Im Φ 1 is the Φ \. This is exactly the abelianized he abelianization of π1 for pathconnected spaces (Allen Hatcher 159) - ϕ\(x\) = \(x, −x\) (Allen Hatcher 159) - ψ\(x, y\) = x + y \. (Allen Hatcher 159) - The exactness of this short exact sequence can be checked as follows (Allen Hatcher 159) - \. (Allen Hatcher 159) - ∂ : Hn \(X\)→Hn−1 \(A ∩ B\) (Allen Hatcher 159) - α ∈ Hn \(X\) is represented by a cycle z (Allen Hatcher 159) - barycentric subdivision (Allen Hatcher 159) - choose z to be a sum x +y of chains in A and B (Allen Hatcher 159) - ∂x = −∂y since ∂\(x + y\) = 0 , (Allen Hatcher 159) - X = S n with A and B the northern and southern hemispheres, (Allen Hatcher 159) - the Klein bottle K as the union of two Möbius bands A and B (Allen Hatcher 159) - The map Φ is twice around Z→Z ⊕ Z , 1 ֏ (Allen Hatcher 160) - 2, −2\) , since the boundary circle of a Möbius band wraps (Allen Hatcher 160) - If only one of f and g , say f , is the identity map, then Z is homeomorphic to what is called the mapping torus of g , the quotient space of X × I under the identifications \(x, 0\) ∼ \(g\(x\), 1\) (Allen Hatcher 160) - C n \(X, A; G\) = Cn \(X; G\)/Cn \(A; G\) (Allen Hatcher 162) - To each category C there is associated a ∆ complex B C called the classifying space of C , whose n simplices are the strings X0 →X1 → ··· →Xn of morphisms in C \. (Allen Hatcher 174) - A natural transformation from a functor F to a functor G induces a homotopy between the induced maps of classifying spaces\. (Allen Hatcher 174) - By regarding loops as singular 1 cycles (Allen Hatcher 175) - the Lefschetz P  number τ\(f \) is defined to be n \(−1\)n tr f∗ : Hn \(X\)→Hn \(X\) \. In particular, if f is the identity, or is homotopic to the identity, then τ\(f \) is the Euler characteristic (Allen Hatcher 188) - χ \(X\) since the trace of the n× n identity matrix is n \. (Allen Hatcher 188) - Theorem 2C\.3\. If X is a finite simplicial complex, or more generally a retract of a finite simplicial complex, and f : X →X is a map with τ\(f \) ≠ 0 , then f has a fixed point\. (Allen Hatcher 188)